Extensive Air Showers Parameters Estimation Using Machine Learning Techniques with Simulations of the FAST Telescope

Technical Summary: Extensive Air Showers Parameters Estimation Using Machine Learning Techniques with Simulations of the FAST Telescope

Problem Statement
The reconstruction of Extensive Air Shower (EAS) parameters from a single Fluorescence detector Array of Single-pixel Telescopes (FAST) unit presents a significant challenge. A single FAST telescope is equipped with only four large photomultipliers (PMTs), resulting in a lack of angular granularity compared to traditional fluorescence detectors. Consequently, distinct shower parameters—such as primary energy, incident geometry, and the depth of shower maximum ($X_{max}$)—can produce similar signals in the PMTs. Traditional "top-down" reconstruction methods, which rely on likelihood functions comparing real data to vast grids of simulations, are computationally expensive and sensitive to initial parameter values. This study investigates whether machine learning (ML) techniques can effectively reconstruct the primary particle energy ($E$) and $X_{max}$ using only the time-structured signal from a single FAST telescope, specifically in a noise-free simulated environment.

Methodology
The study utilizes a dataset of 1,416,236 Monte Carlo simulations generated by the FAST Collaboration. The simulations model proton-induced air showers with primary energies ranging from $10^{17.5}$ to $10^{20.5}$ eV. The $X_{max}$ values are drawn from realistic probability density functions derived from the Conex generator, while other geometric parameters (zenith angle, azimuth, core position) are sampled from uniform distributions within defined physical constraints.

• Data Preprocessing: To ensure data quality, a filtration process was applied to remove trivial signals. Criteria included a maximum amplitude requirement (1–2000 photoelectrons), a removal of signals with narrow temporal widths, and a cut on $X_{max}$ values (600–1000 g/cm²). This resulted in a final dataset of 477,141 events, each consisting of four time-series traces (5,000 time bins each) representing the PMT signals.
• Machine Learning Architectures: Five distinct models were implemented and compared:
  • Model I (Benchmark): A Gradient Boosting Regression (GBR) model.
  • Model II: A feedforward Deep Neural Network (DNN) where all four traces are concatenated into a single 20,000-point input vector.
  • Model III: A DNN that processes the four traces in parallel through separate layers before concatenating the outputs for further processing.
  • Model IV: A 1D Convolutional Neural Network (CNN) treating the concatenated traces as a single-channel image (1 × 20,000).
  • Model V: A 1D CNN treating the four traces as a four-channel image (4 × 5,000).
• Training and Optimization: The dataset was split into training (40%), validation (10%), and testing (50%) subsets. Hyperparameters (number of layers, neurons, kernel sizes) were optimized via grid search using Mean Squared Error (MSE) as the loss function. Normalization was applied to inputs and outputs to the [0, 1] interval.

Key Contributions and Results
The study demonstrates that ML techniques can reconstruct EAS parameters with high precision using data from a single FAST telescope unit, provided noise is controlled.

• Performance Metrics:
  • Energy Reconstruction: The best-performing model (Model III) achieved a Pearson correlation coefficient ($\rho$) of 0.964 between predicted and true $\log_{10}(E/\text{eV})$. The resolution (standard deviation of relative bias) was found to be approximately 0.9%, with a median absolute bias of 0.31%.
  • $X_{max}$ Reconstruction: The correlation for $X_{max}$ was lower, at $\rho \approx 0.757$, with a resolution of approximately 5% (roughly 40 g/cm² for an $X_{max}$ of 850 g/cm²). The relative bias for $X_{max}$ ranged from -8% to +17%, with larger deviations observed in the extreme tails of the distribution where data is sparse.
• Model Comparison:
  • All neural network models (II–V) significantly outperformed the GBR benchmark (Model I), which yielded a correlation of only 0.791 for energy and 0.474 for $X_{max}$.
  • Among the neural networks, fully connected architectures (Models II and III) slightly outperformed the convolutional architectures (Models IV and V). Model III, which processes traces in parallel, was selected as optimal, showing a ~2% improvement in MSE over Model II.
  • The study notes that CNNs exhibited weak periodic structures in their predictions, suggesting that for this specific trace-based inference task, standard dense layers are more suitable than convolutional layers.

Significance and Claims
The authors claim that these results constitute an important verification of the FAST telescope's capabilities in a single-unit configuration. The primary significance lies in demonstrating that sufficient information regarding shower energy and development maximum is encoded in the time structure of the signal from just four PMTs, even without stereo geometry.

The paper positions these ML models as potential tools for:

1. Initial Estimates: Providing rapid pre-classification of events before classical likelihood-based reconstruction.
2. Operational Resilience: Enabling energy spectrum measurements even when parts of a future FAST array are non-operational (e.g., due to weather), although full geometric reconstruction would still require stereo observations.
3. Future Development: Serving as a foundation for more complex configurations, including the planned FAST mini-array.

The authors explicitly state that the current results are based on noise-free simulations. They acknowledge that the next steps involve incorporating realistic electronic noise, night-sky background, and atmospheric non-homogeneities. They caution that while the models perform well as initial estimators, their use as a full replacement for top-down reconstruction in the presence of significant atmospheric non-homogeneities requires further validation. The study concludes that the approach is promising but modestly notes that the current resolution limits are partly due to the limited information inherent in a single telescope configuration.